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ρ[n_, L_]:=
  FixedPoint[
    ArrayFlatten,TensorProduct@@Table[Subscript[ρ, l,m][k],{k,n},{l,L},{m,L}]];

list2[n_, L_]:=
  ParallelTable[ρ[n,L][[i,j]]->a[i,j],{i,L^n},{j,L^n}]/.
   {a[i_,j_]:>ToExpression["a"<>ToString[i]]}//Flatten;

p1[n_, L_] := 
  Sum[
    Subscript[ρ, L, L][k]*
    Product[
     If[k == k1, 1, 
         ParallelSum[Subscript[ρ, l, l][k1], {l, L - 1}]], {k1, n}], {k, n}]//Expand;

P1[n_, L_]:= P1[n,L]= p1[n,L] /. list2[n,L];

p0[n_, L_]:=
  ParallelProduct[
    Sum[Subscript[ρ, l, l][k], {l, L - 1}], {k, n}] // Expand;

P0[n_, L_]:=P0[n,L]= p0[n,L] /. list2[n,L];

Easy to evaluate...

P0[4, 3]-P0[3, 3]
P1[4, 3]-P1[3, 3]

What kind of changes are needed to speed up this calculation...

P0[5, 3]-P0[4, 3]
P1[5, 3]-P1[4, 3]
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up vote 0 down vote accepted

By using the command Rho[n_, L_] := Rho[n, L] = ..., we can make it faster:

Rho[n_, L_] := Rho[n, L] = FixedPoint[
        ArrayFlatten, TensorProduct@@Table[
            Subscript[\[Rho], l, m][k],{k, n}, {l, L}, {m, L}]];

list2[n_, L_] := list2[n, L] = ParallelTable[
        Rho[n, L][[i, j]] -> a[i, j], {i, L^n}, {j, L^n}] /. {a[i_, j_]  
            :> ToExpression["a" <> ToString[i]]} // Flatten;
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